Published on *Department of Mathematics* (https://math.yale.edu)

February 11, 2019 - 4:15pm

TBA

Periodic orbits and geodesics play important roles in the study of dynamics and geometry. Motivated by the classical Prime Number Theorem, it is a natural problem to investigate precise asymptotics of the number of (primitive) periodic orbits and (closed) geodesics, as well as the corresponding error terms. Such results, often known as Prime Orbit Theorems and Prime Geodesic Theorems, have been established in many dynamical and geometric contexts. In this talk, we will discuss recent progresses of Prime Orbit Theorems in complex dynamics.

We will explain our recent work with T.~Zheng on Prime Orbit Theorems for a class of branched covering maps on the $S^2$ called expanding Thurston maps. More precisely, we show that the number of primitive periodic orbits of such maps, under certain H"{o}lder potential with respect to so-called visual metrics on $S^2$, is asymptotically the same as the logarithmic integral, with an exponential error term. Such a result follows from quantitative study of the holomorphic extension properties of the associated dynamical zeta functions and dynamical Dirichlet series, and spectral properties of some carefully defined Ruelle operators. The geometric properties of the visual metrics also play an essential role here. In particular, the above result applies to postcritically-finite rational maps with no periodic critical points.